In the present paper, we use Ramachandra's kernel function of the second order, namely ${\rm Exp} ((\sin z)^2)$, which has some advantages over the earlier kernel ${\rm Exp} (z^{4a+2})$ where $a$ is a positive integer. As an outcome of the new kernel we are able to handle $\Omega$-theorems for error terms in the asymptotic formula for the summatory function of the coefficients of generating functions of the ${\rm Exp}(\zeta(s)), {\rm Exp\,Exp}(\zeta(s))$ and also of the type ${\rm Exp\,Exp}((\zeta(s))^{\frac{1}{2}})$.